CEM: More Bands, Better Performance - Semantic Scholar
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CEM: More Bands, Better Performance Xiurui Geng, Luyan Ji, Kang Sun, and Yongchao Zhao
Abstract—Target detection has recently drawn considerable interest in hyperspectral image processing. People tend to exclude corrupted or badly damaged bands before applying the target detection algorithm to the data for better detection results. In this letter, it is proved that adding any band independent of the original image, even a noisy band, would be always beneficial to the performance of constrained energy minimization in terms of output energy. Finally, several tests are conducted to further justify our viewpoint. Index Terms—Band number, constrained energy minimization (CEM), hyperspectral data, receiver operating characteristic (ROC).
I. I NTRODUCTION
T
ARGET detection is one of the main study regions of remotely sensed data. Because the targets of interest are often small in size or distributed with a low probability and the ground sampling distance of hyperspectral imaging spectrometer is often limited, the targets of interest occur at subpixel levels and mix with other more signatures from the background [1], [2]. In this case, it is hard to detect the targets by making use of the conventional remotely sensed methods. The hyperspectral imaging spectrometer provides the new technique for this problem because of its great number of bands and high spectral resolution. The common hyperspectral detection algorithms include the spectral angle mapping (SAM) [3], spectral feature fitting (SFF) [4], orthogonal subspace projection (OSP) [5]–[7], constrained energy minimization (CEM) [5], [8], matched filter (MF) [1], [9], [10], [11]–[16], and mixture tuned matched filtering (MTMF) [17]–[20]. SAM is to calculate the spectral angle between background and target pixels, and the most advantage is that it can eliminate the influence of illumination difference. However, since SAM has not taken background information into consideration, the results of SAM are usually not that perfect. The SFF method uses spectral absorption feature in its comparisons. It may fail to extract the target of interest when the target does not have obvious spectral characteristics distinguished from those of the background. OSP requires
Manuscript received November 28, 2013; revised January 24, 2014 and March 6, 2014; accepted March 12, 2014. This work was supported by the National High Technology Research and Development Program (863 Program) of China under Grant 2013AA122804. X. Geng and K. Sun are with the Key Laboratory of Technology in Geospatial Information Processing and Application System, Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China. L. Ji is with the Ministry of Education Key Laboratory for Earth System Modeling, Centre for Earth System Science, Tsinghua University, Beijing 100084, China. Y. Zhao is with the Institute of Electronics, Chinese Academy of Sciences, Beijing 100191, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2014.2312319
the spectral signature of both the target and background. It is usually hard for OSP to produce an optimal result in real time. CEM is a linear filter, which constrains a desired target signature while minimizing the total energy of the output of other unknown signatures. CEM requires the spectral prior knowledge of the target and utilizes the second-order statistical information of the image. Under the assumption of lowprobability targets, the CEM detector can distinguish the target of interest from the background very well. MF is another widely used hyperspectral target detection algorithm, which is deduced from the generalized likelihood ratio test [9], [11]. In the Bayes or Neyman–Pearson sense, when the target and background classes follow multivariate normal distributions with the same covariance matrix, the MF detector can get an optimum detection result. In fact, MF and CEM detectors have a similar form, and the main difference is that the data used in MF detector have to be centralized first. MTMF [17], [18] combines the statistical method of the MF with the linear mixing model, which can produce a better detection result. The band generation process (BGP) was presented in [21], which produces new additional bands from the original multispectral bands by multiplying each individual bands by itself, correlating any pair of two different bands, or applying a nonlinear function (such as square-root and logarithm function) to bands. The additional bands generated by the BGP can provide useful information for target detection and classification. Generalized CEM (GCEM) is presented in [22] by incorporating the idea of BGP to expand the original image data so that there are enough spectral band images to make CEM more effective. The results demonstrate that GCEM performed scientifically better than CEM [22]. In this letter, we expect to prove that adding arbitrary bands linearly irrelevant to the original data would be always helpful to achieve lower output energy of CEM. II. CEM The technique of CEM is originally derived from the linearly constrained minimized variance adoptive beam forming in the field of digital signal processing. It used a finite-impulse response (FIR) filter to constrain the desired signature by a specific gain while minimizing the filter output energy [5], [8]. Assume that we are given a finite set of observations S = {x1 , x2 , . . . , xN }, where xi = (xi1 , xi2 , . . . , xiL )T for 1 ≤ i ≤ N is a sample pixel vector, N is the total number of pixels, and L is the number of bands (generally L N ). Suppose that the desired signature d is also known as a priori. The objective of CEM is to design a FIR linear filter w = (w1 , w2 , . . . , wL )T to minimize the filter output power subject to the constraint, dT w = L l=1 dl wl = 1. Then, the problem yields 2 N T = min(wT Rw) min N1 (w x ) i i=1 (1) w
w
dT w = 1
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T where R = (1/N )[ N i=1 xi xi ] turns out to be the sample autocorrelation matrix of S. The solution to (1) is called the CEM operator with weight vector wCEM given by R−1 d (2) wCEM = T −1 . d R d Substitute (2) into (1), and then, we can obtain the objective function as 1 T f (wCEM ) = wCEM (3) RwCEM = T −1 . d R d Equation (3) means that the CEM operator is the optimum linear filter in the sense of output energy. However, we have found and proved that the addition of extra bands can further reduce the filter output energy (1/dT R−1 d) while keeping the gain of the desired signature invariant (dT w = 1). III. I NFLUENCE OF T OTAL BAND N UMBER ON CEM In this section, we will provide the proof that adding bands can increase the performance of CEM. In other words, the performance of CEM will decrease when removing any bands of the data. Suppose that Ω ⊂ (1, 2, . . . , L) is an arbitrary subset of the band index set (1, 2, . . . , L); RΩ and dΩ are the corresponding autocorrelation matrix and target spectral vector, respectively. The theorem is given as follows. Theorem 1: The output energy from full bands is always less than that from the partial bands, i.e., 1 1 < T −1 . (4) T −1 d R d dΩ RΩ dΩ Proof: It is easy to see that, if the theorem holds when Ω = (2, . . . , L), it will hold for any Ω ⊂ (1, 2, . . . , L). Hence, the proof can be changed to prove that inequality (4) holds for Ω = (2, . . . , L). Thus, let Ω = (2, . . . , L), and then, the terms RΩ and dΩ be⎤ ⎡ r22 · · · r2L . . .. .. ⎦ and dΩ = (d2 , d3 , . . . , dL )T , come RΩ = ⎣ .. . rL2 . . . rLL respectively. Next, we rewrite R as ⎤ ⎡ r11 r12 · · · r1L ⎢ r21 r22 · · · r2L ⎥ r11 rT1Ω ⎥ ⎢ R = ⎣ .. (5) .. .. ⎦ = .. r1Ω RΩ . . . . rL1 rL2 · · · rLL and plug it into the denominator of the left term in (4) as T −1 d1 r11 rT1Ω d1 T −1 d R d= . dΩ r1Ω RΩ dΩ
(6)
Then, we expand the inversion of the autocorrelation matrix, R−1 , as −1 r11 rT1Ω −1 R = r1Ω RΩ −1 −1⎤ ⎡ rT r1Ω rT rT r1Ω rT r1Ω 1 1Ω 1Ω 1Ω 1Ω R R + − − − Ω Ω r11 r11 r11 r11 r11 r11 =⎣ −1 −1 ⎦. T r1Ω rT r r 1Ω r 1Ω 1Ω R − RΩ − r111Ω − Ω r11 r11 (7)
Fig. 1. (a) OMIS-II hyperspectral image (band 1: λ = 462.1 nm) and (b) spectral signature of target of interest.
According to the Sherman–Morrison formula [23], [24], the term (RΩ − (r1Ω rT1Ω /r11 ))−1 can be calculated by −1 r1Ω rT1Ω RΩ − r11 = R−1 Ω +
1 1−
−1 1 T r11 r1Ω RΩ r1Ω
1 −1 R r1Ω rT1Ω R−1 Ω . r11 Ω
(8)
Substitute (8) into (7), and we can reduce the inversion of R into −1 r rT1Ω R−1= 11 r1Ω RΩ =
1 r11−rT1Ω R−1 Ω r1Ω
−1 1 −rT1Ω R Ω × −1 −1 −1 −1 . T T −R−1 Ω r1Ω RΩ r11−r1Ω RΩ r1Ω +RΩ r1Ω r1Ω RΩ (9) Furthermore, substitute (9) into (6), and the term dT R−1 d can be simplified as 2 d1 − dTΩ R−1 Ω r1Ω T −1 + dTΩ R−1 (10) d R d= Ω dΩ . r11 − rT1Ω R−1 Ω r1Ω Compare (4) with (10), and it can be found that the inequality (4) holds if r11 − rT1Ω R−1 Ω r1Ω > 0. Hence, we will prove the inequality r11 − rT1Ω R−1 Ω r1Ω > 0 hereinafter. According to the rule for the partitioned matrix determinant, we can have r11 rT 1Ω = r11 − rT1Ω R−1 r1Ω |RΩ |. |R| = (11) Ω r1Ω RΩ Since both R and RΩ are symmetric positive-definite matrices for the general image (we assume that L N holds and R is invertible), the determinants of both R and RΩ satisfy |R| > 0, |RΩ | > 0. Thus, the term r11 − rT1Ω R−1 Ω r1Ω =
|R| > 0. |RΩ |
(12)
Combine (10) with (12), and we can deduce dT R−1 d > dTΩ R−1 Ω dΩ . Thus, (1/dT R−1 d) < (1/dTΩ R−1 Ω dΩ ).
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Fig. 2. Illustration of total band number to the performance of target detection. (a) 4 bands (Ω1 ). (b) 8 bands (Ω2 ). (c) 16 bands (Ω3 ). (d) 32 bands (Ω4 ). (e) 64 bands (Ω5 ).
From Theorem 1, we can conclude that removing any bands from the original data will raise the output energy of the CEM filter. On the contrary, adding any band linearly irrelevant to the original bands will lower the objective function of CEM. Theorem 1 provides the theoretical basis for GCEM’s better results than that of CEM.
IV. E XPERIMENT A. OMIS Data In this section, we verify Theorem 1 presented in this letter by real hyperspectral data [see Fig. 1(a)]. The image has a size of 200 × 200 pixels using OMIS-II (Operational Modular Imaging Spectrometer), which is a hyperspectral imaging system developed by the Shanghai Institute of Technical Physics, Chinese Academy of Sciences. The data are acquired by the Aerial Photogrammetry and Remote Sensing Bureau in Xi’an of China in 2003. The data are composed of 64 bands from visible to thermal infrared (462.1–10 250 nm) with a spatial resolution of 3.6 m. Targets of interest are distributed at two locations around the bottom right corner of the image, occupying about 30 pixels. Fig. 1(b) shows the spectral curves of the target signature. The target signature can be manually selected according to the ground-truth information or automatically determined by endmember extraction methods. In this letter, a fast endmember extraction algorithm, Gaussian elimination method [25], is chosen to extract the target’s spectrum first. In order to evaluate the influence of the total band number to the performance of CEM, we generated five subsets of bands from the original image, noted as Ω1 , Ω2 , Ω3 , Ω4 , and Ω5 . The numbers of bands selected for the five subsets are 4, 8, 16, 32, and 64, respectively, and for each subset, the bands are selected by the variable-number variable-band selection (VNVBS) method [26], respectively. The selected band sets satisfy Ω1 ⊂ Ω2 ⊂ Ω3 ⊂ Ω4 ⊂ Ω5 . It is noteworthy that we use the mean spectral of the data as the reference signature for VNVBS. The detection results using these five subsets are shown in Fig. 2. Visually, the performance of CEM increases as the number of bands increases. The targets of interest cannot be detected at all when only four bands are used [Ω1 , Fig. 2(a)]. However, as the number of bands selected increases from 8 to 64, the contrast between targets and background becomes more and more prominent [see Fig. 2(b)–(e)]. Fig. 3 shows the energy curve of the objective function [see (1) and (3)] as the number of bands we used. As expected, the energy decreases as the number of bands increases. The drop is very sharp when the
Fig. 3. Energy of objective function as function of total band number.
total band number < 16 and tends to be steady when the total band number > 32. The aforementioned experiment confirms the fact that reducing the number of bands will decrease the performance of CEM. On the contrary, if we want to increase CEM’s performance, adding new bands to the original image is undoubtedly a feasible way. Hence, in the next experiments, bands with random noise are added. The generated noise band is uniformly distributed between (0,1) with m = 200 and n = 200. Fig. 4 shows the ten detection results when 1–10 noise bands are added. From the visual perspective, adding more noise bands does not worsen the detection results. The corresponding energy of the objective function is shown in Fig. 5, and again, the output energy of CEM decreases as the number of noise bands increases. From Figs. 4 and 5, we can conclude that adding noise bands does cause reduction in the energy of CEM objective function. However, the more noise bands do not necessarily mean better receiver operating characteristic (ROC) performance. The main reason for this phenomenon is that noise has no extra information for distinguishing the desired target from the background. Therefore, only adding bands with useful information can improve the ROC performance of CEM. That is why GCEM can produce better results when adding nonlinearly generated bands by BGP. In the following, we provide a simple but effective bandaddition method to improve the performance of CEM. Specifically, two additional bands are produced by performing SAM on the original data [see Fig. 6(a)] and the centralized data [see Fig. 6(b)]. Next, these two bands are added to the original data. The comparison of CEM detection results from the original and new data is shown in Fig. 7. It can be seen that the results have a higher contrast between the target and background when two SAM bands are added.
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Fig. 4. Influence of adding noise bands to CEM results. The number of noised bands is increased from one to ten for (a) to (j).
Fig. 5. (a) Output energy as function of total band number when noised bands are added. (b) ROC curves for CEM with different number of noise.
Fig. 6. Detection result of SAM on the (a) original (left) and (b) centralized (right) data.
Fig. 7. Detection results of CEM on the (a) original and (b) new data. (c) Output energy of the two data sets. The new data are generated by adding the two bands in Fig. 6 to the original image.
B. AVIRIS Data In order to further justify Theorem 1, we conduct a new experiment based on another widely used hyperspectral data
Fig. 8. (a) True color of Sandiego data (left) and (b) spectral signature of target of interest (right).
set. The data used in this experiment are acquired by the Airborne Visible Infrared Imaging Spectrometer (AVIRIS) over the Sandiego airport. A subscene [shown in Fig. 8(a)] used in this test consists of 200 × 200 pixels with 224 bands. Among these bands, 22 invalid bands (107–113, 153–166, and 224) are removed. The bands with low SNR are purposefully kept to test the impact of noise bands to CEM results. We choose the three aircrafts presented at the top left of the image as the target of interest to be detected. The spectrum of the target is selected manually as shown in Fig. 8(b). We have compared the results of CEM from three bands sets with different numbers. Specifically, the bands sets are 60 bands selected by VNVBS, 202 valid bands (noisy bands included), and 204 bands which consist of all 202 valid bands and another 2 artificially created bands. These two artificial bands are the results of the spectrum angle mapper and minimum distance, respectively. The detection maps from these band sets are shown in Fig. 9. In order to quantitatively measure the target detection performance of these band sets, we have compared the ROCs [27] of the results (see Fig. 10). It can be seen from the ROC curve that the overall performance of CEM is better when more bands are involved. In particular, when 204 bands are used for CEM, the detection probability is always higher than that from 60 bands for the same false alarm probability. The reason is that, just as the theorem shows, the objective function (output energy) of CEM is decreasing with the increase of involved bands. In addition, the output energy can roughly reflect the ROC performance of CEM, although they are not absolutely equivalent.
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Fig. 9. Detection results from (a) 60 VNVBS-selected bands, (b) 202 total bands, and (c) 204 added bands.
Fig. 10. ROC curve for CEM with different number of bands.
V. C ONCLUSION In this letter, we have proved that more bands are beneficial to the detection of targets with low-probability distribution in terms of output energy. Interestingly, even though we add noisy bands to the original bands, the objective function of CEM does not increase. Moreover, when the additional band contains useful information differentiating the targets of interest from the background, the performance of CEM can be further improved. Thus, how to generate new bands to promote target detection performance is a valuable research direction. R EFERENCES [1] D. Manolakis and G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag., vol. 19, no. 1, pp. 29–43, Jan. 2002. [2] X. R. Geng and Y. C. Zhao, “Principle of small target detection for hyperspectral imagery,” Sci. Chin. Ser. D, Earth Sci., vol. 50, no. 8, pp. 1225–1231, Aug. 2007. [3] F. A. Kruse, A. B. Lefkoff, J. B. Boardman, K. B. Heidebrecht, A. T. Shapiro, P. J. Barloon, and A. F. H. Goetz, “The Spectral Image Processing System (SIPS)—Interactive visualization and analysis of imaging spectrometer data,” Remote Sens. Environ., vol. 44, no. 2/3, pp. 145–163, May/Jun. 1993. [4] R. N. Clark, G. A. Swayze, A. Gallagher, N. Gorelick, and F. Kruse, “Mapping with imaging spectrometer data using the complete band shape least squares algorithm simultaneously fit to multiple spectral features from multiple materials,” in Proc. 3rd Airborne Visible/Infrared Imaging Spectr. Workshop, 1991, pp. 2–3. [5] J. C. Harsanyi, “Detection and classification of subpixel spectral signatures in hyperspectral image sequences,” Ph.D. dissertation, Dept. Elect. Eng., Univ. Maryland, Baltimore, MD, USA, 1993. [6] Q. Du, H. Ren, and C.-I. Chang, “A comparative study for orthogonal subspace projection and constrained energy minimization,” IEEE Trans. Geosci. Remote Sens., vol. 41, no. 6, pp. 1525–1529, Jun. 2003.
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[7] M. K. Arora, S. Bansal, S. Khare, and K. Chauhan, “Comparative assessment of some target detection algorithms for hyperspectral images,” Defence Sci. J., vol. 63, no. 1, pp. 53–62, Jan. 2013. [8] W. H. Farrand and J. C. Harsanyi, “Mapping the distribution of mine tailings in the coeur d’Alene River Valley, Idaho, through the use of a constrained energy minimization technique,” Remote Sens. Environ., vol. 59, no. 1, pp. 64–76, Jan. 1997. [9] D. Manolakis, G. Shaw, and N. Keshava, “Comparative analysis of hyperspectral adaptive matched filter detector,” in Proc. SPIE, 2000, pp. 2–17. [10] J. Y. Chen and I. S. Reed, “A detection algorithm for optical targets in clutter,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-23, no. 1, pp. 46–59, Jan. 1987. [11] D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” in Proc. SPIE, 2009, pp. 733402-1– 733402-16. [12] D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Hyperspectral detection algorithms: Use covariances or subspaces,” in Proc. SPIE, 2009, pp. 74570Q-1–74570Q-8. [13] R. S. DiPietro, D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Performance evaluation of hyperspectral detection algorithms for subpixel objects,” in Proc. SPIE, 2010, p. 76951W. [14] D. Manolakis, D. Marden, J. Kerekes, and G. Shaw, “On the statistics of hyperspectral imaging data,” in Proc. SPIE, 2001, pp. 308–316. [15] C. C. Funk, J. Theiler, D. A. Roberts, and C. C. Borel, “Clustering to improve matched filter detection of weak gas plumes in hyperspectral thermal imagery,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 7, pp. 1410–1420, Jul. 2001. [16] J. Minet, J. Taboury, F. Goudail, M. Péalat, N. Roux, J. Lonnoy, and Y. Ferrec, “Influence of band selection and target estimation error on the performance of the matched filter in hyperspectral imaging,” Appl. Opt., vol. 50, no. 22, pp. 4276–4284, Aug. 2011. [17] J. W. Boardman, “Leveraging the high dimensionality of AVIRIS data for improved sub-pixel target un-mixing and rejection of false positives: Mixture tuned matched filtering,” in Proc. Summaries 7th Annual JPL Airborne Geosci. Workshop, 1998, p. 55. [18] J. W. Boardman and F. A. Kruse, “Analysis of imaging spectrometer data using N-dimensional geometry and a Mixture-Tuned Matched Filtering (MTMF) approach,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 11, pp. 4138–4152, Nov. 2011. [19] J. J. Mitchell and N. F. Glenn, “Subpixel abundance estimates in mixture-tuned matched filtering classifications of leafy spurge (Euphorbia esula L.),” Int. J. Remote Sens., vol. 30, no. 23, pp. 6099– 6119, Nov. 2009. [20] I. Dópido, A. Villa, A. Plaza, and P. Gamba, “A quantitative and comparative assessment of unmixing-based feature extraction techniques for hyperspectral image classification,” IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens., vol. 5, no. 2, pp. 421–435, Apr. 2012. [21] H. Ren and C.-I. Chang, “A generalized orthogonal subspace projection approach to unsupervised multispectral image classification,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 6, pp. 2515–2528, Nov. 2000. [22] C.-I. Chang, J.-M. Liu, B.-C. Chieu, C.-M. Wang, C.-I Chang, C.-W. Yang, D.-J. Ma, C.-S. Lo, and P.-C. Chung, “Generalized constrained energy minimization approach to subpixel target detection for multispectral imagery,” Opt. Eng., vol. 39, no. 5, pp. 1275–1281, May 2000. [23] J. Sherman and W. J. Morrison, “Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix (abstract),” Ann. Math. Stat., vol. 20, p. 621, 1949. [24] J. Sherman and W. J. Morrison, “Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix,” Ann. Math. Stat., vol. 21, pp. 124–127, 1950. [25] X. Geng, Z. Xiao, L. Ji, Y. Zhao, and F. Wang, “A Gaussian elimination based fast endmember extraction algorithm for hyperspectral imagery,” ISPRS J. Photogramm. Remote Sens., vol. 79, pp. 211–218, May 2013. [26] S. Wang and C.-I. Chang, “Variable-number variable-band selection for feature characterization in hyperspectral signatures,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 9, pp. 2979–2992, Sep. 2007. [27] C.-I. Chang, “Target signature-constrained mixed pixel classification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens., vol. 40, no. 5, pp. 1065–1081, May 2002.
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CEM: More Bands, Better Performance Xiurui Geng, Luyan Ji, Kang Sun, and Yongchao Zhao
Abstract—Target detection has recently drawn considerable interest in hyperspectral image processing. People tend to exclude corrupted or badly damaged bands before applying the target detection algorithm to the data for better detection results. In this letter, it is proved that adding any band independent of the original image, even a noisy band, would be always beneficial to the performance of constrained energy minimization in terms of output energy. Finally, several tests are conducted to further justify our viewpoint. Index Terms—Band number, constrained energy minimization (CEM), hyperspectral data, receiver operating characteristic (ROC).
I. I NTRODUCTION
T
ARGET detection is one of the main study regions of remotely sensed data. Because the targets of interest are often small in size or distributed with a low probability and the ground sampling distance of hyperspectral imaging spectrometer is often limited, the targets of interest occur at subpixel levels and mix with other more signatures from the background [1], [2]. In this case, it is hard to detect the targets by making use of the conventional remotely sensed methods. The hyperspectral imaging spectrometer provides the new technique for this problem because of its great number of bands and high spectral resolution. The common hyperspectral detection algorithms include the spectral angle mapping (SAM) [3], spectral feature fitting (SFF) [4], orthogonal subspace projection (OSP) [5]–[7], constrained energy minimization (CEM) [5], [8], matched filter (MF) [1], [9], [10], [11]–[16], and mixture tuned matched filtering (MTMF) [17]–[20]. SAM is to calculate the spectral angle between background and target pixels, and the most advantage is that it can eliminate the influence of illumination difference. However, since SAM has not taken background information into consideration, the results of SAM are usually not that perfect. The SFF method uses spectral absorption feature in its comparisons. It may fail to extract the target of interest when the target does not have obvious spectral characteristics distinguished from those of the background. OSP requires
Manuscript received November 28, 2013; revised January 24, 2014 and March 6, 2014; accepted March 12, 2014. This work was supported by the National High Technology Research and Development Program (863 Program) of China under Grant 2013AA122804. X. Geng and K. Sun are with the Key Laboratory of Technology in Geospatial Information Processing and Application System, Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China. L. Ji is with the Ministry of Education Key Laboratory for Earth System Modeling, Centre for Earth System Science, Tsinghua University, Beijing 100084, China. Y. Zhao is with the Institute of Electronics, Chinese Academy of Sciences, Beijing 100191, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2014.2312319
the spectral signature of both the target and background. It is usually hard for OSP to produce an optimal result in real time. CEM is a linear filter, which constrains a desired target signature while minimizing the total energy of the output of other unknown signatures. CEM requires the spectral prior knowledge of the target and utilizes the second-order statistical information of the image. Under the assumption of lowprobability targets, the CEM detector can distinguish the target of interest from the background very well. MF is another widely used hyperspectral target detection algorithm, which is deduced from the generalized likelihood ratio test [9], [11]. In the Bayes or Neyman–Pearson sense, when the target and background classes follow multivariate normal distributions with the same covariance matrix, the MF detector can get an optimum detection result. In fact, MF and CEM detectors have a similar form, and the main difference is that the data used in MF detector have to be centralized first. MTMF [17], [18] combines the statistical method of the MF with the linear mixing model, which can produce a better detection result. The band generation process (BGP) was presented in [21], which produces new additional bands from the original multispectral bands by multiplying each individual bands by itself, correlating any pair of two different bands, or applying a nonlinear function (such as square-root and logarithm function) to bands. The additional bands generated by the BGP can provide useful information for target detection and classification. Generalized CEM (GCEM) is presented in [22] by incorporating the idea of BGP to expand the original image data so that there are enough spectral band images to make CEM more effective. The results demonstrate that GCEM performed scientifically better than CEM [22]. In this letter, we expect to prove that adding arbitrary bands linearly irrelevant to the original data would be always helpful to achieve lower output energy of CEM. II. CEM The technique of CEM is originally derived from the linearly constrained minimized variance adoptive beam forming in the field of digital signal processing. It used a finite-impulse response (FIR) filter to constrain the desired signature by a specific gain while minimizing the filter output energy [5], [8]. Assume that we are given a finite set of observations S = {x1 , x2 , . . . , xN }, where xi = (xi1 , xi2 , . . . , xiL )T for 1 ≤ i ≤ N is a sample pixel vector, N is the total number of pixels, and L is the number of bands (generally L N ). Suppose that the desired signature d is also known as a priori. The objective of CEM is to design a FIR linear filter w = (w1 , w2 , . . . , wL )T to minimize the filter output power subject to the constraint, dT w = L l=1 dl wl = 1. Then, the problem yields 2 N T = min(wT Rw) min N1 (w x ) i i=1 (1) w
w
dT w = 1
1545-598X © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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T where R = (1/N )[ N i=1 xi xi ] turns out to be the sample autocorrelation matrix of S. The solution to (1) is called the CEM operator with weight vector wCEM given by R−1 d (2) wCEM = T −1 . d R d Substitute (2) into (1), and then, we can obtain the objective function as 1 T f (wCEM ) = wCEM (3) RwCEM = T −1 . d R d Equation (3) means that the CEM operator is the optimum linear filter in the sense of output energy. However, we have found and proved that the addition of extra bands can further reduce the filter output energy (1/dT R−1 d) while keeping the gain of the desired signature invariant (dT w = 1). III. I NFLUENCE OF T OTAL BAND N UMBER ON CEM In this section, we will provide the proof that adding bands can increase the performance of CEM. In other words, the performance of CEM will decrease when removing any bands of the data. Suppose that Ω ⊂ (1, 2, . . . , L) is an arbitrary subset of the band index set (1, 2, . . . , L); RΩ and dΩ are the corresponding autocorrelation matrix and target spectral vector, respectively. The theorem is given as follows. Theorem 1: The output energy from full bands is always less than that from the partial bands, i.e., 1 1 < T −1 . (4) T −1 d R d dΩ RΩ dΩ Proof: It is easy to see that, if the theorem holds when Ω = (2, . . . , L), it will hold for any Ω ⊂ (1, 2, . . . , L). Hence, the proof can be changed to prove that inequality (4) holds for Ω = (2, . . . , L). Thus, let Ω = (2, . . . , L), and then, the terms RΩ and dΩ be⎤ ⎡ r22 · · · r2L . . .. .. ⎦ and dΩ = (d2 , d3 , . . . , dL )T , come RΩ = ⎣ .. . rL2 . . . rLL respectively. Next, we rewrite R as ⎤ ⎡ r11 r12 · · · r1L ⎢ r21 r22 · · · r2L ⎥ r11 rT1Ω ⎥ ⎢ R = ⎣ .. (5) .. .. ⎦ = .. r1Ω RΩ . . . . rL1 rL2 · · · rLL and plug it into the denominator of the left term in (4) as T −1 d1 r11 rT1Ω d1 T −1 d R d= . dΩ r1Ω RΩ dΩ
(6)
Then, we expand the inversion of the autocorrelation matrix, R−1 , as −1 r11 rT1Ω −1 R = r1Ω RΩ −1 −1⎤ ⎡ rT r1Ω rT rT r1Ω rT r1Ω 1 1Ω 1Ω 1Ω 1Ω R R + − − − Ω Ω r11 r11 r11 r11 r11 r11 =⎣ −1 −1 ⎦. T r1Ω rT r r 1Ω r 1Ω 1Ω R − RΩ − r111Ω − Ω r11 r11 (7)
Fig. 1. (a) OMIS-II hyperspectral image (band 1: λ = 462.1 nm) and (b) spectral signature of target of interest.
According to the Sherman–Morrison formula [23], [24], the term (RΩ − (r1Ω rT1Ω /r11 ))−1 can be calculated by −1 r1Ω rT1Ω RΩ − r11 = R−1 Ω +
1 1−
−1 1 T r11 r1Ω RΩ r1Ω
1 −1 R r1Ω rT1Ω R−1 Ω . r11 Ω
(8)
Substitute (8) into (7), and we can reduce the inversion of R into −1 r rT1Ω R−1= 11 r1Ω RΩ =
1 r11−rT1Ω R−1 Ω r1Ω
−1 1 −rT1Ω R Ω × −1 −1 −1 −1 . T T −R−1 Ω r1Ω RΩ r11−r1Ω RΩ r1Ω +RΩ r1Ω r1Ω RΩ (9) Furthermore, substitute (9) into (6), and the term dT R−1 d can be simplified as 2 d1 − dTΩ R−1 Ω r1Ω T −1 + dTΩ R−1 (10) d R d= Ω dΩ . r11 − rT1Ω R−1 Ω r1Ω Compare (4) with (10), and it can be found that the inequality (4) holds if r11 − rT1Ω R−1 Ω r1Ω > 0. Hence, we will prove the inequality r11 − rT1Ω R−1 Ω r1Ω > 0 hereinafter. According to the rule for the partitioned matrix determinant, we can have r11 rT 1Ω = r11 − rT1Ω R−1 r1Ω |RΩ |. |R| = (11) Ω r1Ω RΩ Since both R and RΩ are symmetric positive-definite matrices for the general image (we assume that L N holds and R is invertible), the determinants of both R and RΩ satisfy |R| > 0, |RΩ | > 0. Thus, the term r11 − rT1Ω R−1 Ω r1Ω =
|R| > 0. |RΩ |
(12)
Combine (10) with (12), and we can deduce dT R−1 d > dTΩ R−1 Ω dΩ . Thus, (1/dT R−1 d) < (1/dTΩ R−1 Ω dΩ ).
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Fig. 2. Illustration of total band number to the performance of target detection. (a) 4 bands (Ω1 ). (b) 8 bands (Ω2 ). (c) 16 bands (Ω3 ). (d) 32 bands (Ω4 ). (e) 64 bands (Ω5 ).
From Theorem 1, we can conclude that removing any bands from the original data will raise the output energy of the CEM filter. On the contrary, adding any band linearly irrelevant to the original bands will lower the objective function of CEM. Theorem 1 provides the theoretical basis for GCEM’s better results than that of CEM.
IV. E XPERIMENT A. OMIS Data In this section, we verify Theorem 1 presented in this letter by real hyperspectral data [see Fig. 1(a)]. The image has a size of 200 × 200 pixels using OMIS-II (Operational Modular Imaging Spectrometer), which is a hyperspectral imaging system developed by the Shanghai Institute of Technical Physics, Chinese Academy of Sciences. The data are acquired by the Aerial Photogrammetry and Remote Sensing Bureau in Xi’an of China in 2003. The data are composed of 64 bands from visible to thermal infrared (462.1–10 250 nm) with a spatial resolution of 3.6 m. Targets of interest are distributed at two locations around the bottom right corner of the image, occupying about 30 pixels. Fig. 1(b) shows the spectral curves of the target signature. The target signature can be manually selected according to the ground-truth information or automatically determined by endmember extraction methods. In this letter, a fast endmember extraction algorithm, Gaussian elimination method [25], is chosen to extract the target’s spectrum first. In order to evaluate the influence of the total band number to the performance of CEM, we generated five subsets of bands from the original image, noted as Ω1 , Ω2 , Ω3 , Ω4 , and Ω5 . The numbers of bands selected for the five subsets are 4, 8, 16, 32, and 64, respectively, and for each subset, the bands are selected by the variable-number variable-band selection (VNVBS) method [26], respectively. The selected band sets satisfy Ω1 ⊂ Ω2 ⊂ Ω3 ⊂ Ω4 ⊂ Ω5 . It is noteworthy that we use the mean spectral of the data as the reference signature for VNVBS. The detection results using these five subsets are shown in Fig. 2. Visually, the performance of CEM increases as the number of bands increases. The targets of interest cannot be detected at all when only four bands are used [Ω1 , Fig. 2(a)]. However, as the number of bands selected increases from 8 to 64, the contrast between targets and background becomes more and more prominent [see Fig. 2(b)–(e)]. Fig. 3 shows the energy curve of the objective function [see (1) and (3)] as the number of bands we used. As expected, the energy decreases as the number of bands increases. The drop is very sharp when the
Fig. 3. Energy of objective function as function of total band number.
total band number < 16 and tends to be steady when the total band number > 32. The aforementioned experiment confirms the fact that reducing the number of bands will decrease the performance of CEM. On the contrary, if we want to increase CEM’s performance, adding new bands to the original image is undoubtedly a feasible way. Hence, in the next experiments, bands with random noise are added. The generated noise band is uniformly distributed between (0,1) with m = 200 and n = 200. Fig. 4 shows the ten detection results when 1–10 noise bands are added. From the visual perspective, adding more noise bands does not worsen the detection results. The corresponding energy of the objective function is shown in Fig. 5, and again, the output energy of CEM decreases as the number of noise bands increases. From Figs. 4 and 5, we can conclude that adding noise bands does cause reduction in the energy of CEM objective function. However, the more noise bands do not necessarily mean better receiver operating characteristic (ROC) performance. The main reason for this phenomenon is that noise has no extra information for distinguishing the desired target from the background. Therefore, only adding bands with useful information can improve the ROC performance of CEM. That is why GCEM can produce better results when adding nonlinearly generated bands by BGP. In the following, we provide a simple but effective bandaddition method to improve the performance of CEM. Specifically, two additional bands are produced by performing SAM on the original data [see Fig. 6(a)] and the centralized data [see Fig. 6(b)]. Next, these two bands are added to the original data. The comparison of CEM detection results from the original and new data is shown in Fig. 7. It can be seen that the results have a higher contrast between the target and background when two SAM bands are added.
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Fig. 4. Influence of adding noise bands to CEM results. The number of noised bands is increased from one to ten for (a) to (j).
Fig. 5. (a) Output energy as function of total band number when noised bands are added. (b) ROC curves for CEM with different number of noise.
Fig. 6. Detection result of SAM on the (a) original (left) and (b) centralized (right) data.
Fig. 7. Detection results of CEM on the (a) original and (b) new data. (c) Output energy of the two data sets. The new data are generated by adding the two bands in Fig. 6 to the original image.
B. AVIRIS Data In order to further justify Theorem 1, we conduct a new experiment based on another widely used hyperspectral data
Fig. 8. (a) True color of Sandiego data (left) and (b) spectral signature of target of interest (right).
set. The data used in this experiment are acquired by the Airborne Visible Infrared Imaging Spectrometer (AVIRIS) over the Sandiego airport. A subscene [shown in Fig. 8(a)] used in this test consists of 200 × 200 pixels with 224 bands. Among these bands, 22 invalid bands (107–113, 153–166, and 224) are removed. The bands with low SNR are purposefully kept to test the impact of noise bands to CEM results. We choose the three aircrafts presented at the top left of the image as the target of interest to be detected. The spectrum of the target is selected manually as shown in Fig. 8(b). We have compared the results of CEM from three bands sets with different numbers. Specifically, the bands sets are 60 bands selected by VNVBS, 202 valid bands (noisy bands included), and 204 bands which consist of all 202 valid bands and another 2 artificially created bands. These two artificial bands are the results of the spectrum angle mapper and minimum distance, respectively. The detection maps from these band sets are shown in Fig. 9. In order to quantitatively measure the target detection performance of these band sets, we have compared the ROCs [27] of the results (see Fig. 10). It can be seen from the ROC curve that the overall performance of CEM is better when more bands are involved. In particular, when 204 bands are used for CEM, the detection probability is always higher than that from 60 bands for the same false alarm probability. The reason is that, just as the theorem shows, the objective function (output energy) of CEM is decreasing with the increase of involved bands. In addition, the output energy can roughly reflect the ROC performance of CEM, although they are not absolutely equivalent.
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Fig. 9. Detection results from (a) 60 VNVBS-selected bands, (b) 202 total bands, and (c) 204 added bands.
Fig. 10. ROC curve for CEM with different number of bands.
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